Model experiments with fixed bow ant-pitching fins

/. CHEF

HYDROMEGHANICS

o

AERODYNAMICS

o

STRUCTURAL MECHAN ICS

o

APPLIED MAThEMATICS PRNC-TMe-6i8 (Rev. 3-58)

MCDEL EXPERIMENTS WITH FIXED BOW ANTIPITCHING FINS

by

George P. Stefun

Te.

RESEARCH AND DEVELOPMENT REPORT

December 1959 Report 1118

MODEL EXPERIMENTS _{WITH FIXED BOW} ANTIPITCHING FINS

George P. Stefun

Reprint of Paper Published by The Journal of Ship Research

Volume 3, Number 2 October 1959

Model Experiments With Fixed

Bow Antipitching Fins

By George P. Stefunt

The purpose of this paper is to provide additional information relative to the problem of

designing efficient antipitching devices. The results of seakeeping experiments ore

ere-sented for a model fitted with several alternate designs of fixed bow fins. The bulk of the ¡pejjmnts is concerned with the effects that fins of different aspect ratios have on the

pitching and heaving motions, the phase angles between pitch and heave, the vertical

accelerations, and the speed reductions in waves. Results also are presented showing the feçts on.jtçhin of fin area and of fin-tip fences.

THE effects of fixed bow antipitching fins on the sea-keeping characteristics of ship models have been

investi-gated at the David Taylor Model Basin as part of the

fundamental hydromechanics research program, and for

specific application to individual U. S. naval ships Í 1, 21.2 The present paper gives the results of

experi-ments designed to show the effects of various fin

con-figurations on the pitch and heave, speed loss, phase angles,. and vertical accelerations of a p.6-f t modl in

regular head seas. The test conditions included a speed range corresponding to Fronde numbers from O to 0.30,

and a range of wave lengths corresponding to wave

length-ship length ratios from 0.75 to 1.51.

In general, the results indicate that fixed bow fins

pro-duce maximum Ditch reductions for ship-speed and

wave-length combinations that correspond to near syn-chronous conditions. For the particular models and test conditions of this investigation, maximum Ditch

reductions un to 37 ner cent were obtained with fins of

total plan area eoual to 2.0 per cent of the waterplane

area and aspect ratio eua1 to 2.0. Similar fins with an

I. I ..0 31 ser cent le s effective

A = waterplane area = aspect ratio B = beam CB = block coefficient F = Froude number g = acceleration of gravity

H = draft

Nomenclature h = wave height

K = longitudinal radius of gyration L = length between perpendiculars V = model speed

Z, = amplitude of heave with fins

installed

Zo = amplitude of heave without fins

than the fins of higher aspect ratio, for corresponding"

tests conditions.

The results of a brief study of the effects of fin area on pitch reductions indicate that 2.0 per cent fins weie twice as effective as 1.0 per cent fins with the same aspect ratio. Fins of 4.0 per cent area, however, were only

about three times as effective as the 1.0 per cent fIns. Thus, the increase in effectiveness was directly

propor-tional to the increase in area, only if the areas did not

become too large.

The various fin configurations caused a 0 to 15 ner

cent ncrease in the calm-water resistance of the model. In waves however, this increase was more than offéfi decrease in the motion-induced resistance. In general, therefore, the use of fins resulted in improved ability to maintain speed, especially in synchronous conditions. j

Fin No.

Table i Fin Particulars

Tip

fences Span,

in. Chord,in.

Plan

area Aspectratio (length,width)

5.0 2.5 O.O2A, 2.0 None 2.5 5.0 O.O2A, 0.5 None 3.15 2.1 O.01A,. 1.5 2.1,0.6 4.50 3.0 002A, 1.5 3.0,1.25 4.50 3.0 0.02A,, 1.5 None 6.37 4.25 O.04A,. 1.5 4.25,1.5 2. D'David Taylor Model Basin, Navy Department, Washington,

3. 4.

Numbers in brackets designate references at the end of the 4(a)

paper. 5.

displacement

phase angle by which heave

lags pitch wave length

= amplitude of pitch with fins

installed

Station O

Station 1/2

Station i

Fin Configurations

Experiments were conducted on six different fin in-stallations whose principal characteristics are listed in

r

_{Table 1. All of the fins were simple aluminum plat.e} of

rectangular nlan and cross section. and were mounted on the keel line of the model.

The fins can be divided into two groups as in Fig. 1, which shows their relative dimensions and locations at the bow. The first group consists of Fins i and 2 which had the same areas but different aspect ratios. These fins, with area equal to 2.0 per cent of the waterplane area, A , were not equipped with tipfences. They were mounted forward of the forefoot as indicated in the draw-ings of Fig. i and the photograph, Fig. 2(a). The loca-tion of these fins was impractical (rom the viewpointof structural considerations. It was used here to minimize the interference effects due to hull proximity and to lo-cate the centers of area of both fins at the samedistance from amidships.

The second group consists of Fins 3, 4 and 5 whichhad

the same aspect ratio but different areas. They were fitted with tip fences and installed in a more normal posi-tion with the leading edge located at about the forward perpendicular. The photograph, Fig. 2(b), shows a typical fin installation, and indicates the relative dimen-sions of the fences which were used.

I

A final fin design, designated as Fin 4(a), was obtained

? by constructing fins with the same dimensions as Fin 4

but without fences installed. Test Program and Equipment

The 6.6-ft model which was used in the present study represents a typical naval vessel. A list of model

par-ticulars is given in Table 2. Appendages on the model included rudders and bilge keels.

The experiments were conducted in the TM B-140 ft basin using a gravity towing system for propelling the

1/

Fig. i Planform drawings and fin locations

/ Fir, , , Areo 0.02 Aw

/

/

1F1n3 , oa0.Oi

Station O

Station 1/2

StatiOn 1

model and a pneumatic wavemaker for generating regular waves [3]. The towing system is fixed relative to the wavemaker such that only those headings corresponding to head seas or following seas arepossible. The model was restricted to lead seas in the presentinvestigation.

The model was towed at speeds ranging from O to 2.6

knots in calm water and in waves of 5, 6, 8 and 10 ft length. Wave heights were kept constant at 2.0 in., resulting in wave length-to-wave height ratios of 30 to

60. The wave conditions were purposely intended to be

moderate so that comparisons aniong the various fin

con-figurations would not be influenced by nonlinearities

which usually accompany more severe conditions. The model and test facility were instrumented so that simultaneous measurements were obtained on a Sanborn 8-channel recorder for wave height, pitch androll angles, heave accelerations, and the vertical accelerations attwo points, one forward and one aft of midships. Instrumen-tation included a capacitance-type wave-height probé, a Minneapolis Honeywell vertical gyro, and two Giannini vertical accelerometers. Amplitudes of heave were cal-culated from the amplitudes of vertical acceleration at the

center of gravity. Model speeds were measured using a photocell and electronic counter arrangement at the driv-ing wheel of the gravity towdriv-ing system.

An extensive series of tests was performed on the

model without fins and on the model fittedwith Fins i

Table 2 Model Particulars

Length BP, ft 6 60 Beam, ft 0.862 Draft, ft 0.224 Displacement, lb 46.50 Waterplane area, sq ft 4. 164 Longitudinal gyradius, ft i . 54 Block coefficient 0.61 Waterplane coefficient 0. 73

Natural pitch period 0.70fEC. Area 0.02 A., 2.0 A.R. 1.5 A.R. I Fin5 , Area 0.0k A Fin 2, 0.02

I A. R.= 0.5

z L_ _1

and 2. The results of these tests provide the basis for a general evaluation of the effects on sekeeping of

anti-pitching fins, as well as a specific evaluation of the

effects of the aspect-ratio difference between l'ms i and 2. Experiments with the remaining fin configurations were limited to a few representative test conditions sinca these were sufficient to characterize the effects due to changes in fin areas and tip fences. A summary of the test condi-tions for the various fin configuracondi-tions is given in Table 3.

Pitch and Heave Amplitudes

Curves of pitch and heave amplitudes and the phase angle by which heave lags pitch are plotted against speed

in Figs. 3 through 6 for the model with and without

(a) Fin i located fcrward of Station O

(b) Fin 4 located at Station O

Fig. 2 Typical fin configurations

3

fins. _{The "dashed" portions of the curves represent}

extrapolations to zero speed which were necessary

because of the difficulty of obtaining reliable data at

slow speeds.

In order to facilitate the comparison of the stabilized

and unstabilized motions, faired values from Figs. 3 through 6 are replotted for constant speeds in Figs. 7 and 8 to a base of the wave length-ship length ratio

X/L. Fig. 7 shows the pitch reduction effected by the fins, defined by 100

-

_)/

where is the pitch

amplitude without fins and st', is the corresponding amplitude with fins. _{Similarly, Fig. 8 shows the change} in heave amplitudes in terms of l00(Z - ZF)/ZO where

Zo is the heave amplitude without fins and Z is the

a

-Model Sp..d io OOtnt,

Fig. 3 Motions of model in waves 5 ft long, 2 in. high _ÇY Without Fins - -o-- Fin i )W.R. 2.0)

A.-- Fin 2 (A.R. 0.5)

Table 3

The curves of Fig. 7 indicate that the greatest nitc reductions occur for wave-length and speed combinations

which result in periods of encounter that are near the

model's natural pitch period, in other words, the fins

attain their maximum effectiveness at

synchronous conditions. The curve for 2.4 knots speed, for example,

reaches a maximum at X/L = 1.05 and then declines

rapidly for longer wave lengths. The curves for very slow speeds never reach a peak value since the

synchro-nous wave lengths corresponding to these speeds are

below the range of this investigation.

The curves of Fig. 8 indicate that the fins can have the

e t sí in reasin the heave motion es.eciall

in the longer waves. In general, the curves show an

increase in heave at all speeds in waves longer than

1.4L, and heave reductions at all speeds in waves shorter than 0.75 L. In waves of intermediate length, the fins cause an increase in heave at low speeds and a decrease at high speeds.

4

80

Test Conditions

=::

Model mood in tonte

Fig. 4 Motions of model in waxes 6 ft long, 2 in. high

in-60 in-60 io to

---R H ---R

r- .:i

Without F n,

-- --e---- Fini (A.R.02.h

-

0

Fin 2 (A.5. O0.0)

08

0'_

to

Vertical Accelerations

Amplitudes of vertical accelerations are plotted versus speed in Figs. 3 through 12 for the several test conditions. One accelerometer was located at a point 18.5 in. forward of the center of gravity of the model, and another was 18.5 in. aft. The algebraic sum of the two signals was obtained electrically to provide a record of the acceleration at the position of the center of gravity.

The three records give the vertical

accelerations at positions which correspond approximately to the

mid-point (station 10) and to the two quarter mid-points (stations 5 and 15) of the model length. The results indicate that the amplitudes of accelerations at various points on the model are generally reduced through the use of fins atthe bow by an amount proportional to themotion reduction. The largest accelerations, which occur at the ends of a ship, are reduced in proportion to the amount of pitch

reduction since for these points, the pitching motion contributes a major component of the vertical

acceler-Without fins Fins 1 and 2 Fins 3, 4a, 5 Fin 4 0-2 .6 0-2.6 0-2.6 0-2.6 5,68, 5,6,8, 5 5,6 0)1 10 3 0.75-1.51 0.75-1.51 0 75 0.75,0.91 Speed runge, \Vave h,

knots length, ft in. X/h

0

:

.'

r

--

/

Model Speed in Knut,

Fig. 5 Motions of model in waves 8 ft long, 2 in. high

t-,

ations. For positions ìear or somewhat aft of midship, however, the use of fins can sometimes cause an increase

ii _{the accelerations, corresponding to an increase in the}

heave component of the motion. These accelerations

rnall enough so that any increase due to fins may slot be important except in cases svhere ship instrumentsor

other equipment may require locations iìear points of

minimum motion.

Resistance

The model resistance in calm water and in waves is plotted in Figs. 13, 14 and 15 in terms of the ow force required to obtain a given speed. A comparison of Fig. 13 (without fins) with Figs. 14 and 15 (with fins)

indi-cates that the fins generally increase the resistance in

calm water and in waves longer than about l2Oper cent

..thjpJegth. The resistance in waves with lengths

between 75 and 120 per cent of the ship length, however, is. 1ss thap that without fins. The improved resistance

characteristics can he attributed tu the motion

reduc-tions effected by the fins since the lower resistance meas-urements were obtained for test conditions which cor-respond roughly to the region of maximum motion reduc-tion, Figs. 7 and 8.

Faired values from the resistance curves of Figs. 13, 14 and 15 are replotted in Fig. 16 for the model with and without fins. The speed in waves corresponding to a

5

4

O Without Fin,

- -o_--AFin i U.S. 2.0)

Fin i (A.0.oO.5)

20

$05.1 Op,.5 in Knot.

Fig. 6 Motions of model in waves 10 ft Long, 2 in. high

given calm-water tow force is plotted against the _wave length-ship length ratio A/L. The curves show a maxi-mum speed reduction for wave lengths of about 90 per centofthe ship length. The amountofspeed loss due to waves, relative to the calm-water speed, was consistently less for the fin-equipped model except forvery long waves. It is, of course, realized that the rectangular_cross sec-tioiis and the keel-line locations of the fins used in _this

investigation are not suitable for optimum resistance

characteristics, especially in calm water. _{However, this}

'as little beariii on the maor obsrv

.,

.1 ' , a

j-which result from the resistance measurements, since these are based on the relative performance of various fins of similar shape and orientation anl on the relative performance of fins in calm ivater and in waves. Also, differences in resistance or in speed-maintaining ability are a function mainly of differences in motion characteristics, which are little affected by shape of fin

sections or by minor variations in fin alignment. Fin Aspect Ratio

A comparison of the pitch-amplitude results obtained for Fins 1 and 2 Fig. 7, indicates that a large change in fin aspect ratió has a significant effect on the amount of

pitch reduction that can be accomplished.

_{For the}

particular model and test conditions whichwere used in this investigation, ns of

aspect ratio 0.5 were roly

-O---- Without Fin.

- -0--- Fin i A.R.2.0)

tu - A--- Fin 2 (A.M. 0.5)

hAIR1

u

za

Io

20

004 O 2

M (

Wave Length in Feet

s , 8 9 SYNCHSONISM Fin I. (A.R 2.0 V (Knot 24 s) 03 Z 4

Wave Length to Ship Length Ratio

Fig.7 Pitch reductions effected by fins I and 2

6

0 32

424

Wave Lenith in Feet

i - 1 -Fin 2 (A.R 00.5) k V 2. 2 0 Knots VQ Fin i (A.R. = 2.0) Y = 2.' Knots

ì!'È1ì

0.8

P1L

SYS1CHRONISM i I ¡ I lin 2 (A.R.4- 0.5)

!i!1I!I

Wittoos4t Firo Fin12 (A(1. Fin. B.. 0.. 0.5)2.0)

-o--

-44 -44-.

#1R

IO15 _.-4 ZI_.__ STATuS 5 -.---O--- Without Pio Fi.i2 (0.0.(AB. Fin. - 2.0) 0.5)

--

-0 ___

ii1_

STATICS 10 STATION15

- - - - .--4 n

Froude tu.b.r 7s4. Iluabor o.' 08 0

Ii

4 I

Wave Length to ship Length Ratio

Fig. 8 Heave reductions effected by fins iand Z

Nod.i S..d in toots 24

Nodet Sp..d in 4-8000,

Fig. 9 Accelerations of model in waves 5 ft long, 2 in.high Fig. 10 Accelerations of model in waves 6 ft long, 2

in. high 04 24 20 -to 40 io to

Rl o,, e Po,.d. 8ob.O 24 804.1 Sp..R InOnoto 7 Foo4. Nob.,'

Fig. 11 Accelerations of model in waves 8 ft long, 2 in. high Fig. 12 Accelerations of model in waves loft long, 2 in. high

30 per cent less effective in reducing pitch amplitudestm

than were similar fins of aspect ratio 2.0. It should

be notedhat the increased effectiveness was obtained b a 300 per cent increase in the value for aspect ratio.

Small changes in fin dimensions that leave the fin

area unaltered, therefore, can be expected to have

negligible effects on pitch reductions.

As indicated in Fig. 16, n increase in fin aspect ratio also improved the resistance characteristics of the model both in calm water and in waves. In calm water, where the bare hull resistance was increased by the addition of

fins at the bow, the fins of aspect ratio Q caused a lesser increase than did those of aspect ratio 0.5. In the

present case, the increases were about 10 and 15 per

cent respectively. For wave lengths of the order of the ship length, fins improved the resistance of the model as compared with that for the bare-hull condition. The improvement due to the higher aspect ratio fins were somewhat greater than that due to those of lower aspect ratio. For wave lengths corresponding to X/L = 1.5, fins again caused an increase in resistance, but differences due to aspect ratio tended to disappear. Judging from

the results obtained for a large change in fin aspect

ratio, minor variations in fin dimensions should have a negligible effect on resistance or sneed in waves.

J

Fin Area

The fins designated as 3, 4 and 5 were tested under a single wave condition consisting of a wave length equal

0

$0

STATOO# 5

a,iI

Pio Pb, Wi0000t i (6.8. 2 )A.R. PIO. = 2.0) 0.5) -O -.8 #8 STOTIOO 10 - ST6TIOX 15 -08

-

-

0 ===', 2"s0AT)o ₅ -PIo Pin Withont 1 (6.8. 2 )A.8. Pin, 2.0) 0.5) - -0'- -e.--0150208 10 O

--0160108 15

t.---

.

-Wan. I H.jght 2.0 Inoh.. Cal. W.t.r W... 1.08th- 5 W.,. I..oth' 6 Foot W b,o8th 8 F..t 8i.n8th = 10 P..t I O

- o

_{- a}

o.-murn

Mi'

IIUIIIII__

_{V'U__}

pr,

í

2 0. 20 za

Nodal Spo.d lo [not.

08 z

'

RO 24

0005.1 Sp..d In [not.

Fig. 13 Resistance of model without fins

R,

0g

Model Speed in Knot, o 32 4 8 e, 02 3.0 IW.V.0eint:2.0l:0h98 I I W.,.Length' 5Feet Wane L.ngth- 6 Feet Wove LenCth 8 F..t

Wove L.ngthiO Feet

1Ua

J.0

- --

- ô

- -

-- --e r

/1

UUUUIUUUU

/

UUMWIUUU

__URU

U.

R...

_pr

pi

W.V.Height -I I 2.0 Inch.. COl. Water

UOVOLength = 5 ie.t 40V.L.egth 6 Hect Mono L.ogth B He.

W... L.ngthli Feet

-- 0

-

_{- - n}

-

RI.-.-_iiii

_rAi...

«iIi1I

0

lIlI.

/

I I I' I I I

Wave Height 2.0 lOche.

UI.UUUUUU

1thotF1c

2.0) To. Forre o80 lb

UUIURUIIUUUIIURUU

-

IRURURI

II... .I.UUUUI.

URUUUI.IUUUU

RRRIIIIURUIIIUUU

IIIIÏRUUUUURRUURUU

U UIUUUIiUUU

_{I UI.IUIpIUI.I.}

Fig. 14 Rcsistance of model with fin 4R..2 Fig. 15 Resistance of model with fin 2

Ror

Weve Length in Feet

.3? 06 o5 0 2

wove Length to Ship Length Rotin

Fig. 16 Speed reductions for constant tow force

04 08 4 28

Model Sp.ed in lÇevte

08 24 29

09

08

03 02

s 40 rrood. 14,b.o

1

Nòd.1 31,0.4 in1400t4 04

Fig. 17 Pitch amplitudes and pitch reductions in waves 5 ft long, 2 in. high

to 75 per cent of the ship length, and a wave length to

height ratio of A speed range corresponding to

Froude numbers of Lt 0.3 was obtained. All these

fins had an aspect ratio of 1.5 but had areasof,jnd4

per cent of the waterplane area, respectively. All were equipped with tip fences and were mounted on the keel

with the leading edge located at about the forward

perpendicular.

The effects of increased fin area on pitch amplitudes are given by the curves in Fig. 17. Pitch amplitudes as well as pitch reductions are plotted against speed for the model with and without fins. As would be expected, the

I results indicate that fins of larger area produce larger pitch reductions. However, for these specific test con-ditions, fins with an area greater than 2per cent of the

waterplane area did not produce reductions in direct

proDortion to the increase in area. For example, while th2..per cent fin was twice as effective as the 1.0 uer çert

..fin the 4 per_cent fin wss nnlyabout three.times

,s

effeçtjye even though it had four times the area4. Test conditions corresponding to longer wave lengths were not included in the present investigation, but the trends with

respect to fin areas should be roughly similar in long

waves to those obtained in the short wave length. The resi.stance curves for Fin 4 in calm water and in waves are presented in Fig. 18. A comparison of this figure with Figs. 14 and 15 indicates that the resistance results obtained for the "normal" configuration of Fin 4 are entirely similar to those obtained for the "special" configurations of Fins i and 2. Also, the pitch

redue-9 z 04 a7 06 0-s 0.4 0.) al

Fig. 18 Resistance of model with fin 4

tions of Fins 3, 4 and 5, Fig. 17, show the same trends as those of Fins i and 2, Figs. 3 and 7, for corresponding test conditions. Accordingly, the conclusions obtained

from the more complete investigation of Fins i und 2 can be extended to "normal" fin configurations. In

other words the results indicate that for normal fin loQations, information with respect to conditions f9r

maximum pitch reductions, or minimum speed loss in

wave8. for example, can be inferred frqm

tha.,J-tamed for fins mounted forward of the forefoot.

Quanti-tative data also can be obtained for normal locations,

but these will be simple approximations. For example, the tests indicate 28 per cent pitch reduction for Fin 4 at 2 knots in 5-ft waves, compared with 31 per cent for Fin i under the same conditions. It is reasonable to expect,

therefore, that Fin 4 should show about 33 per cent reduction at 2.4 knots in 7-ft waves since the tests indicate 37 per cent reduction for Fin 1 under these

conditions.

Tip Fences

Vertical end plates, commonly called tip fences, werg" installed on the fins used in the studies of references [1)

and [2]. Their use was based on a consideration of

elementary flow principles which indicate that the plates

should help eliminate vorticity about the fin tips ajil

-I WOO.H.i1ht 61122 I FinPto. . - ' f

-O---Withont .0 -A Vi.-0.91

ru

0' 1 W. C.to

/

¿'pr,

-- .

---0-O -,0__ Fin Pin P1,5 Withont 3 (A.. 1. (AO.. (An....%) Fin. -1%)

--

.

-04 06 IL 20 24 24 1400.1 5p..d in [.ota

Proud. puab.r

Yd.1 Sp..d In I00t

Fig. 9 Pitch amplitudes and pitch reductions for fins 4and 4a

thus increase the effective aspect ratio and area of the

jj

observations from model tests have indicated

that fences tend to reduce somewhat the adverse flow searation effects that are experienced when the fins have large amplitudes of motion [2].

Fin 4(a) was tested in waves of length equal to 75 per cent of the ship length in order to obtain anestimate of

the effects of tip fences on pitch reduction, from a

comparison with the results obtained for Fin 4. Fin

4(a) with iR 00 1.5 and area = 0.02 A was exactly

the same as Fin 4 with fences removed. The results of

the comparison are presented in Fig. 19, where pitch

amplitudes and pitch reductions are plotted against

speed for both fins. I1h curves indicated that the use of fences nrovided an additional pitch reduction of about 5 per cent above that obtainedfor the fins without fences. Further experiments are needed to determinethe effects on pitching of fences that are cpnsiderably smaller than the large fences used in the presentinvestigation.

Future Research

The experimental results presented in this paper were intended to provide primarily, much neededinformation for the design of efficient antipitchingfins. They were

also to serve, however, as the basis for evaluating and

lo

improving existing theoÑtical methods. It was believed, for example, that an investigation oflarge changes in fin

aspect ratio, thus effecting large changes in fin-lift

characteristics, should provide a valid basis for evaluating the hydrodynamic-lift theory of antipitching fins. It soon became apparent, however,that firm conclusions could not be reached without a more accurateknowlede of. among other things, the lift) drag, and inertia char-acteristics of oscillating hydrofoils. The necessary fundamental research has not been accomplishedat TMB as yet because efforts have been directed toward problems

of more immediate consequence. The experimental

results are published here with the hope that others will

be stimulated and aided in theirwork of a more basic

nature.

One of the problems currently being investigated at

TMB is that of transverse hull vibrations associated with antipitching fins. The study is directed toward determining the nature and cause ofadverse fin-induced vibrations, and the corrective measures which may be necessary to eliminate or reduce these vibrations. In addition, the study wjll provide furtherinformation with respect to the effectiveness for pitch reductions of a large variety of fin shapes, locations,and orientations.

Other work which is planned for future research

includes an investigation of the effectiveness of anti-pitching fins under confused-seas conditions. Such a

study is necessary in order that predictions based on

regular-wave tests can be accepted with greater confi-dence. In addition, the results should be of value for model and full-scale correlations.

Acknowledgments

Some of the experimental results presented in this paper were obtained by Mr. S. E. Lee, whose comments and suggestions throughout the course of the

investiga-tion are also greatly appreciated.

The author is

in-debted also to Mr. J. Bonilla-Noratwho helped with the

testing and data analysis and to Mrs. F. Poole who

helped with fhe data reductionand computations.

References

i U. A. Pournaras, "Pitch Reductionwith Fixed Bow Fins on a Model of the Series 60, 0.60 Block Coefficient,"

DTMB Report No. 1061, October 1956.

2 U. A. Pournaras, "Sea Behavior of a

Mariner-Class Ship Equipped with Bow Anti-Pitching Fins,"

DTMB Report No. 1084, October 1958.

3 F. H. Todd, "Resistance Tests and Motion

Ob-servations on Models in a Seaway," TMB Report No. 931, November 1954.

4 V. G. Szebehely, "Apparent Pitching Axis," Forschungsheftfür Schifftethnik, 1956.

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I Dir, Fluid Mech Lab, Colombia Ocie,

New York

I Dir, Fluid Mech Lab, Univ of Cahtornia,

Berke ley

Dir, Davidsoo Lab, SIT, Mobnken

i Dr. B. V. Korvin-Kroukovsky Dir, Exp Nao Tank, Unjo of Michigan

Ann Arbor

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1 Dir, Inst for Fluid Dyn A AppI Math, Univ of Maryland, College Park

1 Din, Hydrau Lab, Uiriv of Colorado, Boulder

1 Dir, Hydrau Res Lab, Univ of Connecticot, Storno

1 Dir, Scripps Iot of Oceanography, Univ of California, La Jolla

Dir, Fluid Mech Lab, New Yonh Univ. New York Oir, Robinson Hydrau Lab, Ohio St Univ.

Columbus

1 Dir, Hydrau Lab, Peno St Univ. University Park

i Dir, OR L, Peon So 1mm, Universily Park

i Dir, Woods Moie Oceanographic lost, Woods Hole

1 Dir, i'iydnau Lab, Uriv of Wisconsin, Madison 1 Dir, Hydrau Lab, Univ nf Washington, Seattle

1 Admin, Webb Inst nf Nao Arch, Cien Cnve

2 Dir, Iowa Inst of Hydrau Reo, St Univ of Iowa, Iowa City

I Dr. L. Londwvben

1 Din. SI. Anthovy Falls Hydrau Lab, Unie nf

Minnesota, Minneapolis

1 Din of Res, Tech Inst. Northwestern Univ. Evanston

3 Head, Dept NAME, MIT, Cambridge

1 Prof, M. A. Abkowitz

1 Prof. F. M. Lewis

1 Editor. Bibliography of Tech Rpts, OTS, US Dept uf

Commence

3 NNSB & OD Co., Nevont News i Assi Nao Arch

2 Dir Hydrau Lab

i _{Janres Forrestal Res Cli, Library, Princeton Univ.} Princeton, Atto: Mr. M. H. Smith, Assi to Dir

1 Tech Dir, Ship Struc Corn, NRC. Washington

I Dr. M. L. Albertson, Head, Fluid Mech Res, Dept of Civil Engin, Colorado St Unìv, Font Collins

1 Mr. J. P. Breslin, Tech Din, Davidson Lab, SIT, Floboken

i Dr. V. G. Szebehely, Genl 01cc Co, Aeon Sciences Lab, Philadelphia

I _{Dr. A. Andreoni, Tech Res Irst, Eop Ship Model Tank,}

Cama PostaI 7141, Sao Paolo, Brazil

i Prof. C. W. Prohaska, Ship Model Basin, 'djontekaersoej 99,

Klampeobong, Deneark

i Din, Hydra Lab, NRC, Ottawa, Canada

1 _{Chief Supt, Nao Res Estab, Halitan, Nova Scotia, Canada}

I Prof. L. Howarth, Dept. of Math, Univ of Bristol, Bristol, England

2 Dim, AEW, vaslan, Gasport, liants, England

i Mr. E. C. Topper

U ALOSNA, London, England

1 Dir, BORA, 5, Cheoterfield Gardens, Curzoo St, London, W. 1, England

I _{Oir, Ship Div, NFL, Teddington, Middlesen, England} I RADM R. Brord, Din, Bassin d'Essais des Carenes,

6 Blvd Victor, Paris (15e), Fnance

i Dr. L. Malavand, Office Nati d'Etudes et de Recherches P.eronaofiques, 25 Ave de la Drvisinn-LeClerc ChatitlonSaus-Bagneua (Seine), Paris, France

11

Dr. J. Dreodonne, Dir Gen, Inst de Recherches de la Canstruc Navale, 47 Pue dv Monreau, Paris (9e), France

Dr. Siegfried Schuster, Head, Berlin Model Basin, Berlin 87, Germany

Dr. H. W. Lembo. Dir, Hamburg Model Basin. Hamburg 33

Ge r many

1 Dr. Georg Weinbium, Universilaet Hamburg, Berliner Tor 21,

Hamburg, Germany

I _{Gen, Ing. U. Pugliese, Pmo, Instituto Nazionale per Studi}

ed Esperienze di Arch Nao via della Vasca Navale US, Pone, Italy

O _{Dr. J. Okabe. Bes Inst (sr AppI Mech, Kyoshu Unie,}

Hokozaki-Machi, Fukunka-shi, Japan

i Mr. J. Germiisma, Delfi Shipbidg Lab, Prof. Mekelweg, Oeiff, The Netherlands

2 Dir, Nederlandsch Scheepsbouwkundig Proefotafion, Haagsteng 2, Wageningen, The Netherlands

1 Or. G. Vossers

O Prof. J. K. Lorde, Skipsmodelltankev, Trondheim,

Norway

i Senor Manuel Lapez-Acevedo, Dir, Caoal de Eopeniencias Ridrodinamicas, El Pardo (Madrid), Spain

On. Hans Edstrand, Dir, Statens Skeppspnovoingsanntalt,